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A137338 Triangle read by rows: T(n,k), 0 <= k <= n, gives the coefficients of the Charlier polynomials (with parameter a=1), ordered by rising powers.

%I A137338 #33 Oct 28 2019 02:26:12
%S A137338 1,-1,1,0,-3,1,3,6,-6,1,-12,-9,26,-10,1,45,3,-109,71,-15,1,-198,81,
%T A137338 501,-475,155,-21,1,1071,-786,-2663,3329,-1455,295,-28,1,-6984,6711,
%U A137338 16510,-25495,13729,-3647,511,-36,1,53217,-60309,-117912,216004,-135961,43897,-7994,826,-45,1,-462330,589197,953711
%N A137338 Triangle read by rows: T(n,k), 0 <= k <= n, gives the coefficients of the Charlier polynomials (with parameter a=1), ordered by rising powers.
%C A137338 Row sums are 1, 0, -2, 4, -4, -4, 44, -236, 1300, -8276, 61484, etc.
%C A137338 Matrix inverse is A216916. - _Peter Luschny_, Sep 21 2012
%H A137338 Carl V. L. Charlier, <a href="https://babel.hathitrust.org/cgi/pt?id=chi.73230316&amp;view=1up&amp;seq=103">Über die Darstellung willkürlicher Funktionen</a>, Arkiv För Matematik, Astronomi Och Fysik, Band 2, No. 20 (Meddelande från Lunds Astronomiska Observatorium, Series I, No. 27), 1905, 1-35. [Accessible only in the USA via the <a href="https://www.hathitrust.org/accessibility">HathiTrust Digital Library</a>.]
%H A137338 M. Dunster, <a href="https://doi.org/10.1006/jath.2001.3595"> Uniform asymptotic expansions for Charlier polynomials</a>, J. Approx. Theory, 112 (2001), pp. 93-133.
%H A137338 Chun-Hua Ou, <a href="http://math.nist.gov/~DLozier/SF21/SF21slides/Ou.pdf">Global Asymptotics of the Charlier Polynomials via the Riemann-Hilbert Method</a>.
%H A137338 Wikipedia, <a href="https://en.wikipedia.org/wiki/Carl_Charlier">Carl Charlier</a>.
%H A137338 Wikipedia, <a href="https://en.wikipedia.org/wiki/Charlier_polynomials">Charlier polynomials</a>.
%F A137338 Charlier polynomials: C_{n}(a; x) = Sum_{k=0..n} binomial(n,k)*binomial(x,k)*k!*(-a)^(n-k).
%e A137338 [0]     1,
%e A137338 [1]    -1,      1,
%e A137338 [2]     0,     -3,       1,
%e A137338 [3]     3,      6,      -6,      1,
%e A137338 [4]   -12,     -9,      26,    -10,       1,
%e A137338 [5]    45,      3,    -109,     71,     -15,     1,
%e A137338 [6]  -198,     81,     501,   -475,     155,   -21,     1,
%e A137338 [7]  1071,   -786,   -2663,   3329,   -1455,   295,   -28,   1,
%e A137338 [8] -6984,   6711,   16510, -25495,   13729, -3647,   511, -36,   1,
%e A137338 [9] 53217, -60309, -117912, 216004, -135961, 43897, -7994, 826, -45, 1.
%p A137338 with(PolynomialTools):
%p A137338 C := (n, x) -> if n>0 then expand((x-n)*C(n-1,x)-n*C(n-2,x))
%p A137338 elif n = 0 then 1 else 0 fi:
%p A137338 A137338_row := n -> CoefficientList(C(n,x), x);
%p A137338 for n from 0 to 7 do A137338_row(n) od;
%p A137338 # _Peter Luschny_, Sep 21 2012
%t A137338 Ca[x, -1] = 0; Ca[x, 0] = 1; Ca[x_, n_] := Ca[x, n] = (x - (n - 1) - 1)*Ca[x, n - 1] - n*Ca[x, n - 2]; Table[ExpandAll[Ca[x, n]], {n, 0, 10}]; a = Table[CoefficientList[Ca[x, n], x], {n, 0, 10}]; Flatten[a]
%Y A137338 Cf. A216916.
%K A137338 tabl,sign
%O A137338 1,5
%A A137338 _Roger L. Bagula_, Apr 07 2008
%E A137338 Edited by _Peter Luschny_, Sep 21 2012